Maximum scattered linear sets and MRD-codes

The rank of a scattered -linear set of , rn even, is at most rn / 2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r, n, q (rn even) for scattered -linear sets of rank rn / 2. In this paper, we prove that the bound rn / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered -linear sets of of maximum rank n yield -linear MRD-codes with dimension 2n and minimum distance . We generalize this result and show that scattered -linear sets of of maximum rank rn / 2 yield -linear MRD-codes with dimension rn and minimum distance n - 1

A new family of maximum scattered linear sets in PG(1,q^6)

We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of PG(1; qn)" (2019) to a more general family, proving that such linear sets are maximum scattered when q is odd and, apart from a special case, they are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of PG(1; qn)" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters (6; 6; q; 5)

A new family of MRD-codes

We introduce a family of linear sets of PG(1,q^2n) arising from maximum scattered linear sets of pseudoregulus type of PG(3,q^n). For n=3,4 and for certain values of the parameters we show that these linear sets of PG(1,q^2n) are maximum scattered and they yield new MRD-codes with parameters (6,6,q;5) for q>2 and with parameters (8,8,q;7) for q odd

A new family of maximum scattered linear sets in PG(1,q6)\mathrm{PG}(1,q^6)

We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019) to a more general family, proving that such linear sets are maximum scattered when $q$ is odd and, apart from a special case, they are are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters $(6,6,q;5)$

Classes and equivalence of linear sets in PG(1,q^n)

The equivalence problem of GF(q)-linear sets of rank n of PG(1,q^n) is investigated, also in terms of the associated variety, projecting configurations,GF(q)-linear blocking sets of Rédei type and MRD-codes. We call an GF(q)-linear set L_U of rank n in PG(W,Fq^n) = PG(1,q^n) simple if for any n-dimensional GF(q)-subspace V of W, L_V is PGammaL(2, q^n)-equivalent to L_U only when U and V lie on the same orbit of GammaL(2,q^n). We prove that U = {(x,Tr(x)) : x \in GF(q^n)} defines a simple GF(q)-linear set for each n. We provide examples of non-simple linear sets not of pseudoregulus type for n > 4 and we prove that all GF(q)-linear sets of rank 4 are simple in PG(1,q^4)

Scattered trinomials of Fq6[X]\mathbb{F}_{q^6}[X] in even characteristic

In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [B. Csajb\'ok, G. Marino and F. Zullo: New maximum scattered linear sets of the projective line, Finite Fields Appl. 54 (2018), 133-150; G. Marino, M. Montanucci and F. Zullo: MRD-codes arising from the trinomial $x^q+x^{q^3}+cx^{q^5}\in\mathbb{F}_{q^6}[x]$, Linear Algebra Appl. 591 (2020), 99-114], the authors proved that the trinomial $f_c(X)=X^{q}+X^{q^{3}}+cX^{q^{5}}$ of $\mathbb{F}_{q^6}[X]$ is scattered under the assumptions that $q$ is odd and $c^2+c=1$. They also explicitly observed that this is false when $q$ is even. In this paper, we provide a different set of conditions on $c$ for which this trinomial is scattered in the case of even $q$. Using tools of algebraic geometry in positive characteristic, we show that when $q$ is even and sufficiently large, there are roughly $q^3$ elements $c \in \mathbb{F}_{q^6}$ such that $f_{c}(X)$ is scattered. Also, we prove that the corresponding MRD-codes and $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not equivalent to the previously known ones

Vertex properties of maximum scattered linear sets of PG(1,qn)\mathrm{PG}(1,q^n)

In this paper we investigate the geometric properties of the configuration consisting of a $k$-subspace $\Gamma$ and a canonical subgeometry $\Sigma$ in $\mathrm{PG}(n-1,q^n)$, with $\Gamma\cap\Sigma=\emptyset$. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of $\Sigma$ from the vertex $\Gamma$. In particular we deal with the maximum scattered linear sets of the line $\mathrm{PG}(1,q^n)$ found by Lunardon and Polverino and recently generalized by Sheekey. Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajb\'ok and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in $\mathrm{PG}(1,q^6)$, yielding also to new examples of MRD-codes in $\mathbb F_q^{6\times 6}$ with left idealiser isomorphic to $\mathbb F_{q^6}$